High quality audio speakers are typically designed to operate best over a limited range of frequencies. Consequently, crossover filters or networks have long been used in audio systems to separate the band of audio frequencies into two or more sub-bands, with each sub-band used to drive a separate speaker. Desired characteristics of crossover network include relatively flat response, rapid roll off at the cutoff frequency or frequencies, minimum phase response and a minimum number of components.
The following notation will be used herein with respect to the various equations which are presented:                fc is the cutoff frequency (Hz);        fs is the sampling rate (Hz);        ωc is the angular cutoff frequency;        ωc=2 Pi*fc is the angular cutoff frequency in continuous time;        ωc=2 Pi*fc/fs is the angular cutoff frequency in discrete time;        s is the s transform parameter;        z is the z transform parameter;        zi=1/z: this is implemented as a delay element in discrete time systems;        zt=zi/(1−zi): this is implemented as an integrator in discrete time systems.        
As is known, a two-way crossover network comprises a low pass filter and a high pass filter (a network for more than two speakers would include one or more intermediate band pass filters). Numerous types of crossover networks have been developed, each with its own transfer function and resulting characteristics. Butterworth, Tchebychev and Bessel filters are among the most widely used. In addition, crossover networks may be implemented in different “orders”. A first order network is relatively simple, has in-phase outputs and has a roll off of 6 dB/octave. Because there is significant output beyond the crossover frequency, the speaker drivers must be able to handle the corresponding energy.
A second order network, such as the popular Butterworth, is more complex but, as illustrated in the frequency response plots of FIGS. 1A and 1B, the low pass and high pass elements have sharper roll offs of 12 dB/octave (reducing the demand on the drivers); however, the outputs of the low pass and high pass filters are 180° out of phase, as illustrated in the phase response plots of FIGS. 1C and 1D. In the FIGs, the cutoff frequency fc is 10 Hz. The transfer functions of the low pass and high pass continuous time second order Butterworth filters are:
                    BW2_LP        =                                            ω              c              4                                                          (                              s                +                                                                            (                                              -                        1                                            )                                                              1                      /                      4                                                        ⁢                                      ω                    c                                                              )                        ⁢                          (                              s                -                                                                            (                                              -                        1                                            )                                                              3                      /                      4                                                        ⁢                                      ω                    c                                                              )                                                          (        1        )                                BW2_HP        =                              s            2                                              (                              s                +                                                                            (                                              -                        1                                            )                                                              1                      /                      4                                                        ⁢                                      ω                    c                                                              )                        ⁢                          (                              s                -                                                                            (                                              -                        1                                            )                                                              3                      /                      4                                                        ⁢                                      ω                    c                                                              )                                                          (        2        )            As illustrated in the plots of FIGS. 2A and 2B, the low pass filter has 2 complex conjugate poles, while the high pass filter has the same poles as the low pass filter plus two zeroes at 0. A second order Linkwitz-Riley network may be designed from combining second order low pass and high pass Butterworth filters and has the following combined transfer function:
                              LR          ⁢                                          ⁢          2                =                                            s              2                                                      (                                  s                  +                                                                                    (                                                  -                          1                                                )                                                                    1                        /                        4                                                              ⁢                                          ω                      c                                                                      )                            ⁢                              (                                  s                  -                                                                                    (                                                  -                          1                                                )                                                                    3                        /                        4                                                              ⁢                                          ω                      c                                                                      )                                              +                                                    ω                c                4                                                                    (                                  s                  +                                                                                    (                                                  -                          1                                                )                                                                    1                        /                        4                                                              ⁢                                          ω                      c                                                                      )                            ⁢                              (                                  s                  -                                                                                    (                                                  -                          1                                                )                                                                    3                        /                        4                                                              ⁢                                          ω                      c                                                                      )                                                                        (        3        )            FIGS. 3A and 3B are plots of the amplitude and phase responses, respectively, of a second order Linkwitz-Riley network (showing plots of the low pass component, the high pass component and their sum), again having a cutoff frequency of 10 Hz.
A fourth order Butterworth network is still more complex and has an even sharper roll off of 24 dB/octave. This network is generally not economically feasible to implement as a passive network.
A fourth order Linkwitz-Riley network, which is typically designed from two series-connected second order Butterworth filters, retains the sharp roll off advantage of fourth order filters and has the added advantages of having a substantially flat frequency response and having outputs which are 6 dB down at the crossover frequency (instead of only 3 dB for other filters) and in-phase. A fourth order Linkwitz-Riley network may be designed by cascading two second order Linkwitz-Riley networks as follows:
                              LR          ⁢                                          ⁢          4                =                              LR4_HP            +            LR4_LP                    =                                                    s                4                                                                                  (                                          s                      +                                                                                                    (                                                          -                              1                                                        )                                                                                1                            /                            4                                                                          ⁢                                                  ω                          c                                                                                      )                                    2                                ⁢                                                      (                                          s                      -                                                                                                    (                                                          -                              1                                                        )                                                                                3                            /                            4                                                                          ⁢                                                  ω                          c                                                                                      )                                    2                                                      +                                          ω                c                4                                                                                  (                                          s                      +                                                                                                    (                                                          -                              1                                                        )                                                                                1                            /                            4                                                                          ⁢                                                  ω                          c                                                                                      )                                    2                                ⁢                                                      (                                          s                      -                                                                                                    (                                                          -                              1                                                        )                                                                                3                            /                            4                                                                          ⁢                                                  ω                          c                                                                                      )                                    2                                                                                        (        4        )            FIGS. 4A and 4B are amplitude and phase response plots of a continuous time, fourth order Linkwitz-Riley network (showing plots of the low pass component, the high pass component and their sum).
For digital audio signals, the above-described crossover networks may have digital counterparts. A particular digital implementation of a fourth order Linkwitz-Riley filter is designed as a cascade of second order filters with programmable coefficients. In order for the cutoff frequency to be selectable over a reasonable range, such as 30-300 Hz, with low distortion, the filter coefficients must have very high accuracy. FIG. 5 illustrates one stage of such a filter 100. Two such stages would be combined for the low pass section of the crossover network and two more such stages would be combined for the high pass section. The full design requires 16 state variables, 20 multipliers with 20 coefficients and 16 adders. Moreover, in some applications in which the audio stream is highly over-sampled, a first decimation stage may be required, adding to the complexity.
Consequently, a need remains for a high quality crossover network having an easily implemented design with a minimum number of operations, coefficients and state variables and which does not require a decimation stage.